An M × M covariance matrix R exhibits many special properties. For example, it is complex Hermitian, equal to its conjugate transpose, RH = R; it is positive semidefinite, xHRx ≥ 0. Because of the latter, its eigenvalues are greater than or equal to zero as well. In many cases, it is also Toeplitz Ri,j = Ri + m,j + m, that is, diagonal elements are equal. In this chapter, I will review some of the more important properties of covariance matrices and their eigenstructure, and discuss some simple applications.